Types of Discontinuity

Points of discontinuity for $f\left(x\right)=\left[x^2\right] x\in \left[1,2\right]$ 

Let $f$ be a composite function of $x$ defined by $f\left(u\right)=\frac{1}{u^3-6u^2+11u-6}$ where, $u\left(x\right)=\frac{1}{x}$. Then the number of points $x$, where $f$ is discontinuous is:

In $\left(0,5\right)$, the functions $\left[\sin x\right]+\left[\sin 2x\right]$ discontinuous at

The function $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\sin \left(\ln \left|x\right|\right)& ;x\ne 0\end{array}\\ \begin{array}{cc}1& ;x=0\end{array}\end{array}\right.$

The jump of the function $f\left(x\right)=\frac{1-a^{{\displaystyle \frac{1}{x}}}}{1+a^{{\displaystyle \frac{1}{x}}}},a>0$ at its point of discontinuity is 

The function $f\left(x\right)={\tan }^{-1}\left(\frac{1}{x-5}\right)$ has:

If $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\left[\cos \pi x\right], & x\le 1\end{array}\\ \begin{array}{cc}\left|2x-3\right|\left[x^2-2\right],& x>1\end{array}\end{array}\right.\begin{array}{}\\ \end{array}$

Then number of points of discontinuity of $f\left(x\right)$ in $x\in \left[0,2\right]$ is.

The number of points at which the function $f\left(x\right)=\frac{1}{x-\left[x\right]}$ is not continuous is 

A discontinuous function $y=f\left(x\right)$ satisfying $x^2+y^2=4$ is given by $f\left(x\right)=\dots$

Let $f:\left[0,2\right]\to \mathrm{ℝ}$ be defined by $f\left(x\right)=\left[x\right]-\left[x+\frac{1}{2}\right]$, where $\left[x\right]$ denotes the greatest integer less than equal to $x$. At how many points of $\left[0,2\right]$, is $f$ discontinuous ?

If [.] denotes the Greatest integer function, then $f\left(x\right)=[x{]}^2-\left[x^2\right]$ is discontinuous at

The function $f\left(x\right)=\frac{4-x^2}{4x-x^3}$ is :

Let $g\left(x\right)=\left\{\begin{array}{c}1-x;\\ x+2;\\ 4-x;\end{array}\right.\begin{array}{c}0\le x\le 1\\ 1<x<2\\ 2\le x\le 4\end{array}$, then the numbers of points where $g\left(g\left(x\right)\right)$ is discontinuous is

$$\begin{gathered} f\left(x\right)=\frac{\sin (a+x)+\sin (a-x)-2\sin a}{x\sin x} \\ =\sin a \end{gathered}$$ $\left.\begin{array}{l}\text{ for }x\ne a\\ \text{ for }x=a\end{array}\right\}$ is Discontinuous at $x=a$

$f\left(x\right)=\left\{\begin{array}{l}\frac{{\left(3^{\sin x}-1\right)}^2}{x\log (1+x)} \mathrm{for} x\ne 0\\ 2\log 3 \mathrm{for} x=0\end{array}\right.$is Discontinuous at $x=0$.

Discuss the continuity of the following functions. Which of these functions have removable discontinuity? Redefine such a function at the given point so as to remove discontinuity.

$$\begin{gathered} f\left(x\right)=\frac{\sin (a+x)+\sin (a-x)-2\sin a}{x\sin x} \\ =\sin a \end{gathered}$$ $\left.\begin{array}{l}\text{ for }x\ne a\\ \text{ for }x=a\end{array}\right\}$ at $x=a$

Discuss the continuity of the following functions. Which of these functions have removable discontinuity? Redefine such a function at the given point so as to remove discontinuity.

 $f\left(x\right)=\left\{\begin{array}{l}\frac{\left|x\right|}{x}\\ 1\end{array}\right.\begin{array}{l}\text{ for }x\ne 0\\ \text{ for }x=0\end{array}$ at origin

Discuss the continuity of the function. Is this functions has removable discontinuity ?.  Redefine such a function at the given point so as to remove discontinuity.

$f\left(x\right)=\left\{\begin{array}{l}\frac{{\left(3^{\sin x}-1\right)}^2}{x\log (1+x)} \mathrm{for} x\ne 0\\ 2\log 3 \mathrm{for} x=0\end{array}\right.$,at $x=0$

Discuss the continuity of the function $f\left(x\right)$ defined by 

$f\left(x\right)=\begin{array}{ll}\frac{4^x-e^x}{6^x-1},& \mathrm{for} x\ne 0\\ \log \left(\frac{2}{3}\right),& \mathrm{for} x=0\end{array}\} \mathrm{at} x=0$

If the discontinuity is removable redefine the function, so that it becomes continuous.

Discuss the continuity of the function $f\left(x\right)$ defined by $f\left(x\right)=\begin{array}{ll}\frac{1-\sin x}{{\left(\mathrm{\pi }-2\mathrm{x}\right)}^2},& \mathrm{for} x\ne \frac{\mathrm{\pi }}{2}\\ \frac{2}{7},& \mathrm{for} x=\frac{\mathrm{\pi }}{2}\end{array}\} \mathrm{at} x=\frac{\mathrm{\pi }}{2}$

If the discontinuity is removable redefine the function, so that it becomes continuous.